## Optimized Windows

We close this chapter with a general discussion of*optimal windows*in a wider sense. We generally desire

(4.59) |

but the nature of this approximation is typically determined by characteristics of audio perception. Best results are usually obtained by formulating this as an

*FIR filter design problem*(see Chapter 4). In general, both time-domain and frequency-domain specifications are needed. (Recall the potentially problematic impulses in the Dolph-Chebyshev window shown in Fig.3.33 when its length was long and ripple level was high). Equivalently, both

*magnitude*and

*phase*specifications are necessary in the frequency domain. A window transform can generally be regarded as the frequency response of a

*lowpass filter*having a

*stop band*corresponding to the side lobes and a

*pass band*corresponding to the main lobe (or central section of the main lobe). Optimal lowpass filters require a

*transition region*from the pass band to the stop band. For spectrum analysis windows, it is natural to define the

*entire main lobe*as ``transition region.'' That is, the pass-band width is zero. Alternatively, the pass-band could be allowed to have a finite width, allowing some amount of ``ripple'' in the pass band; in this case, the pass-band ripple will normally be maximum at the main-lobe midpoint ( , say), and at the pass-band edges ( ). By embedding the window design problem within the more general problem of FIR digital filter design, a plethora of optimal design techniques can be brought to bear [204,258,14,176,218].

### Optimal Windows for Audio Coding

Recently, numerically optimized windows have been developed by Dolby which achieve the following objectives:- Narrow the window in time
- Smooth the onset and decay in time
- Reduce side lobes below the
*worst-case masking threshold*

### General Rule

There is rarely a closed form expression for the optimal window in practice. The most important task is to formulate an*ideal error criterion*. Given the right error criterion, it is usually straightforward to minimize it numerically with respect to the window samples .

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Window Design by Linear Programming

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Gaussian Window and Transform